=Paper=
{{Paper
|id=Vol-1638/Paper67
|storemode=property
|title=Transferable utility distribution algorithm for multicriteria control in strongly coupled system with priorities
|pdfUrl=https://ceur-ws.org/Vol-1638/Paper67.pdf
|volume=Vol-1638
|authors=Mikhail I. Geraskin
}}
==Transferable utility distribution algorithm for multicriteria control in strongly coupled system with priorities ==
https://ceur-ws.org/Vol-1638/Paper67.pdf
Mathematical Modeling
TRANSFERABLE UTILITY DISTRIBUTION
ALGORITHM FOR MULTICRITERIA CONTROL IN
STRONGLY COUPLED SYSTEM WITH PRIORITIES
M.I. Geraskin
Samara National Research University, Samara, Russia
Abstract. The problem of multicriteria control on the basis of aggregate utility
maximizing is considered. The system utility distribution algorithm in case of
unequal criteria set by priority vector, has been developed for strongly coupled
technical and organized systems with transferable utility. The Pareto efficiency
is proved for the distribution generated on the algorithm basis.
Keywords: multicriteria control, aggregate utility, strongly coupled system,
transferable utility, distribution algorithm, criteria priorities
Citation: Geraskin MI. Transferable utility distribution algorithm for mul-
ticriteria control in strongly coupled system with priorities. CEUR Workshop
Proceedings, 2016; 1638: 542-551. DOI: 10.18287/1613-0073-2016-1638-542-
551
Introduction
For technical and organizational systems there are now two ways to solve the problem
of multicriteria optimal control, efficiency of which is evaluated with the help of sev-
eral criteria. The first way is based on the Pareto efficiency criterion [1] with setting-
up of a corresponding set. Another way is Neumann-Morgenstern expected utility
maximization [2] in one form or another [3-6]. The latter approach ensures the
uniqueness of the solution of the problem. However, its use is justified in case the
strongly coupled system model is adequate [7] for the particular group of criteria. The
system is regarded as strongly coupled in case the criteria of its elements are compati-
ble with the axioms [8] of preference (reflexivity, connection, transitivity, etc.). It
allows for the criteria vector the reduction towards the scalar problem of aggregate
utility optimization. Solution of this problem is Pareto-efficient [10] in case it corre-
sponds to the minimax criterion [9]. Utility aggregation may be based on physical and
technical properties of systems. In particular, in the terminal control problems [11]
deviations from the phase coordinates target values are aggregatively set out in the
form of a metric (distance) in the criteria hyperspace. In the problems of multicriteria
statistical estimation [12] the moments of random variables are set out in the from of
the variation coefficient. In general, the criteria aggregation is based on the properties
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of the system integrity and hierarchal pattern [13], for example, interrelated technolo-
gies implementing systems. Due to the production process uniformity the elements of
criteria are reduced to the aggregated criterion of the system.
The special case of a strongly coupled system is the transferable utility system with
criteria expressed in a single measuring element, for example, in case of homogene-
ous technologies [14] performance optimization. In general any system of criteria may
be formally reduced to the transferable system by means of setting out [9] criteria in a
non-dimensional form by means of corresponding normalization. Transferable utility
allows for the system elements criteria vector values redistribution (transfer), corre-
sponding to a certain Pareto-efficient control. Consequently, the problem of mul-
ticriteria control is reduced to the problem of optimal utility distribution between the
system elements on the basis of the aggregated criterion. Solution of the latter prob-
lem will also be Pareto-efficient.
Transferable utility distribution algorithms are intended for the systems with elements
having equisignificant criteria (of anonymous elements). In this case the aggregation
process doesn't take the criteria priorities into account. In particular, Pareto-efficiency
is justified [15] in case of an algorithm that establishes the minimum between the
element optimum and average non-distributed utility of the system. In case the system
elements criteria have different priorities, the distribution algorithms are reduced to
the median multicriteria choice [16]. However, in a general Pareto- inefficient case,
mechanisms of anonymous symmetric coalitions [17] have been developed. In particu-
lar cases these mechanisms are Pareto-efficient. We shall hereinafter consider the
problem of development of a Pareto-efficient algorithm of transferable utility distribu-
tion between elements with priorities on the basis of the results [14], obtained for
anonymous elements.
1 Problem definition
Let us consider the system with the elements efficiency determined by the criteria
vector f f k u , k K . Let the scalar optimum u k0 for the criteria of k-element be
determined on the basis of the condition
u k0 Arg max f k u , f k0 f k uk0 , k K , (1)
u k U k
where u – control, U – feasible region, f k u , k K – k-element efficiency criteri-
on.
Let us form the criteria minimums vector, required for further normalization, from the
values obtained in case of scalar optimums of other elements:
jK \ k
f kmin min f k u 0j , k K . (2)
Subject to (1), (2) the range of criteria values variation equals to f k u k f k0 , f kmin .
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Let us introduce the hypothesis of benevolence [7], in accordance to which the ele-
ments of a strongly coupled system maximize the utility of the whole system. Control
is determined on the basis of the maximum value of the sum of criteria in case of
optimums of elements. In this context the unique solution of a multicriteria problem is
formed. This solution is Pareto-efficient according to the Herneyer's theorem [1,9]. Let
us denote this solution by u * and define it as follows:
u * Arg max max f j uk , f k* f k u * , k K , (3)
jK k k
kK u U
where f k* – k-criterion value in case of the strongly coupled system optimum.
Let us define the strongly coupled system maximum aggregate utility F u * in case
of control (3) as exceeding of criteria optimums over their minimum values:
F u * f k* f kmin . (4)
k K
Let us introduce the system utility distribution vector X xk , k K , belonging to
the acceptable set
FX X x k , k K : x k F u * , X Rk .
kK
Let us denote the criteria vector, formed as the result of aggregate utility distribution
f k xk , and, with account taken of (4), let us define it as follows:
f k xk f kmin xk , k K . (5)
Criteria functions (5) constitute the efficiency criteria of utility distribution between
elements in case of Pareto-efficient control (3). Let us formulate the multicriteria
problem of optimal utility distribution as follows:
X * Arg max f k x k , k K . (6)
X FX
Let us introduce the criteria priorities vector k , k K , the components of which de-
termine their relative significance within the framework of the system aggregate utili-
ty. As hereinafter the problem (6) is formulated as minimization of criteria deviations
from their scalar maximums, it is expedient to pass on to inverse priorities
k 1 k , k K , that characterize the relevant significance of other criteria in
comparison with this particular criteria. Let us define the vector of inverse priorities
as follows
k R , k K , k 1 . (7)
kK
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As it follows from (1), (3) that deviations of criteria (5) from scalar maximums (1) are
negative, let us define the normalized values of criteria f k xk , k as squared relevant
~
deviations of criteria (5) and (1) with account taken of inverse priorities:
f x f k0
2
f k xk , k k k mink
~
0
, k K. (8)
k fk fk
Firstly, normalization (8) reduces criteria to the range f k xk , k 0,1 that makes it
~
possible to operate on the criteria (8) as transferable ones; secondly, it prescribes the
utility transitivity f xk , k f xi , k k i , k K \ i in respect of which higher prior-
~ ~
ity criteria have lesser utility loss.
Let us consider a strongly coupled system, in which control is defined according to
(3). It also ensures utility transferability with the help of (8). It allows for [7] the re-
duction of the multicriteria problem (6) towards the scalar optimization problem by
means of introduction of an aggregated utility function in the form of the relative
departures product (8):
X * Arg min f k xk , k .
~ (9)
X FX kK
Criterion in the problem (9) constitutes an aggregated function of the system utility.
Moreover, as it is in a normalized form (8) that partial criteria components form part
of (9), the effect distribution that satisfies (9) provides [9] with the maximum efficient
solution of the multicriteria problem (6) in respect of minimax
f X min max f k xk , k ,
~* * ~
X FX kK
that corresponds, as it is shown in [10], to Pareto efficiency.
2 Utility distribution algorithm
Let us formulate the system (4) utility distribution that is optimal in respect of criteria
(9) in the form of the following assertion: distribution
f f 0i k f i*0i
xk*
1
k
F u * f i* 0 k
i
,k K
(10)
1 iK \ k iK \ k
iK \ k i
is the solution of the problem (9).
Proof: differentiating Lagrange function, written down for the problem (9) with con-
sideration of (8)
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f x f k0
2
L f k xk , k F xk k k mink
~
F xk ,
0
kK kK
kK k kf f k kK
we arrive at the system of necessary conditions of optimality
k f k xk f k0 f x f 0, k K 0 2
L/xk 2
f min
f
f f
0 2
iK \ k
i i i
min
k
0 2
(11)
k k k i i k
L/ F xk 0 . (12)
k K
Eliminating Lagrange multiplier from (11), we get
k f k xk f k0 f x f f x f f x f , k , n K , n k ,
0 2 0 0 2
i
i i i n n n n i i i i
f k k
min
f f f f f f f
k
0 2
iK \ k
i i
min
i
0 2
i n
min
n
0 2
iK \ n
i i
min
i
0 2
whence we arrive at the system
f x f f x f f x f f x f , k , n K , n k ,
k k k k
0
i i i i
0 2
n n n n
0
i i i i
0 2
iK \ k iK \ n
or
k f k xk f k0 n f n xn f n0 , n k K (13)
Inserting system utility (4) and utility of elements (5) into (13) we arrive at the distri-
bution (10). Differentiating (11), we shall get the system of sufficient conditions of
optimality
L
// f x f 0, k K ,
i i i i
0 2
xk
f f
iK \ k k i
min
i
0 2
that are met in case of any fi xi , fi 0 , fi min .
As distribution (10) maximizes the aggregate criterion (9), consequently, in accord-
ance with the results of [10], it is Pareto-efficient.
On the basis of the solution (10) of the multicriteria problem (9) let us lay down the
aggregated utility distribution algorithm in the form of the series of the following
steps.
1. Input of the given data: functions of criteria f f k u , k K and restrictions de-
fining the ranges of acceptable controls Uk.
2. Determining scalar optimums of criteria (1) and minimums of criteria (2).
3. Determining optimums of criteria in the system (3) and utility of the system 4 (4).
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4. Defining the priorities vector (7).
5. Calculating the utility distribution (10) and the optimal criteria vector (5).
The algorithm realization requires for the criteria feasible region to be not empty
U i U k , k K \ i and also for the functions f f k u , k K to have finite
maximums in corresponding feasible regions U k . These conditions are met in practi-
cal problems, because as a rule the regions U k , are compact and usually they coincide
for different criteria of one system.
3 Utility optimal distributions modelling
Modelling was done in the context of the three-element system of electronics market,
the elements of which are retailer, bank, insurer, with the functions f k u , k K
having the form
f k u k p k u kbk 1 c k u k , k K ,
where f k , u k , c k – k-element utility (profit), output and marginal costs, p k , bk –price
trends parameters on corresponding markets. Scalar optimums of elements (1) were
defined according to the formulas:
1
ck bk
u 0
,k K
.
pk bk 1
k
Parameters of the model, criteria (1) scalar optimums, criteria (2) minimums, criteria
(3) system optimums are given in Table 1; system (4) utility equaled F=6989. Calcu-
lation of parameters of the model is given [14, 18] for the electronics retailer LLC
“Eldorado”, insurer OJSC “Insurance company “Renaissance”” and bank LLC
“Home Credit and Finance Bank”, that were assigned indices k=1,2,3 corresponding-
ly. In the considered system element 1 gets maximum utility (utility leader), parame-
ter u10 for which corresponds to the volume of sales of goods and is expressed in phys-
ical terms, and utility obtained by elements 2,3 (utility outsiders), are significantly
lower. However, their participation is required for the system to be integral. That is
why the system utility distribution problem becomes. Outputs u20 , u30 correspond to
the volume of insurance and lending and are expressed in monetary terms.
Let us suppose that elements may assign their priorities themselves in the course of
integration into the system, following two variants of strategic behaviour.
Firstly, lets consider the variant of non-cooperative game [19], elements wherein don't
form coalitions, they assign priorities simultaneously and independently. In this case
the Cournot game model [20], symbolized by the letter «K», has the form
k K arg max k , k K and leads to the priorities vector k K *K , k K . The
k 1
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resulting distribution x k* K , k K (fig. 1) corresponds to the elements equisignifi-
cance. In this case the element 1 gets the predominant share of distributed utility, and
the share of utility distributed in favour of the elements 2,3 is reduced concurrently
with the growth of the equilibrium values of priorities *K . Model of game with the
institutional leader sensu Stackelberg [21], symbolized by the letter «S», takes the
k 1
form k S arg max k , l S arg max l , k K \ l , where l-element is the leader;
k l 1
the model results in the vector of priorities k S *S , l*S *S , k K \ l . Distribu-
tion x k* S , k K (fig. 1) shows that the leader sensu Stackelberg (l=3) gets the ad-
vantage in spite of being the utility outsider.
Table 1. Parameters of the model and criteria vector
Parameter of the Element
model 1 2 3
bk -0,09 -0,19 -0,165
ck 22000 0,06 0,025
pk 50000 0,780 0,350
u k0 3210 240548 2962123
f k0 6985, 3,4 14,6
0
f k* 6985, 3,0 12,0
0
f kmin 7,0 1,2 2,4
f f
k
*
k
min
6978, 1,8 9,6
0
x
8000
7000
6000
5000
4000
3000
2000
1000
0
-1000
-2000
0 0,05 0,1 0,15 0,2 0,25 0,3 γ
x*1(K) x*2(K) x*3(K)
x*1(S) x*2(S) x*3(S)
Fig. 1. Utility distribution structure in case of non-cooperative behaviour
1K 2K 3K, S
1
2 ,
S S
3
1
S
,
1
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Secondly, let us consider the variant of cooperative game [22] symbolized with the
letter “S”. In this case elements may form coalitions. The game model of the form
k C arg max k , l C arg max l , l L , k K \ L , where index l L desig-
k 1 k l 1
nates the elements of coalition, leads to the vector of priorities of the form
l C *C , *C k*C , k K \ l . Distribution x k*C , k K (fig. 2) shows that in
comparison with non-cooperative behaviour (fig.1), the coalition of elements (1,2)
significantly increases the utility leader's utility and increases the outsider's utility
insubstantially. Coalition of utility outsiders (2,3) leads to a drastic utility redistribu-
tion in their favour.
x
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
-1000 0 0,1 0,2 0,3 0,4 0,5 0,6 γ
x*1(С),(1,2) x*2(С),(1,2) x*3(С),(1,2)
x*1(С),(2,3) x*2(С),(2,3) x*3(С),(2,3)
Fig. 2. Utility distribution structure in case of cooperative behaviour
1
C 1
,
2
,
2
C 1
,
2
3
C
,1
,
2
1 2
C 2
,
3 3
C 2
,
3 1
C 2
,
3
In all the cases examined the system utility was fully distributed between the ele-
ments. This confirms Pareto efficiency of distributions that are formed on the
basis of the developed algorithm. Modelling showed that the developed algorithm
allows to analyze the impact of the priorities vector on the distribution structure
and to build the vector of the priorities that express different aims of the system
integration.
Conclusion
The algorithm of solving multicriteria problems of control in strongly coupled sys-
tems has been developed. This algorithm allows to determine the optimal system
aggregated utility distribution in case of inequisignificant efficiency criteria on the
basis of the minimum of deviations of criteria from scalar optimums. The algorithm
application sphere includes technical and organizational systems, efficiency criteria of
which are quantifiable and preference relations of which have been determined. In
case the system elements utilities are transferable, which is in general ensured by the
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Mathematical Modeling Geraskin MI. Transferable…
criteria normalization. The algorithm is based on the normalized criteria aggregation
and in case of the utility function optimization with account taken of priorities, it
leads to the minimax distribution (of the guaranteed result). Consequently, the algo-
rithm determines the Pareto-efficient solution of the multicriteria problem.
The use of algorithm allows to reduce the multicriteria problem of optimization in the
control space to the set of scalar problems of optimal control and the problem of mul-
ticriteria distribution in the criteria space. As the result the unique solution is estab-
lished, and, moreover, solving of the multicriteria problem is significantly simplified.
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